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The substitution method is a powerful tool for solving inequalities and equations in algebra. It involves replacing variables with specific values to determine if the inequality or equation holds true. In this exercise, we substitute the point (-1,3) into each inequality one at a time.

For example, for inequality (a) \(x - 3y > 6\), substitute \(x = -1\) and \(y = 3\). The inequality becomes \(-1 - 3(3) > 6\) which simplifies to \(-10 > 6\). This is false, meaning the point does not satisfy the inequality.

By systematically substituting the point into each inequality, students can clearly see which inequalities the point satisfies and which it does not. This is especially beneficial for more complex problems involving multiple variables.

Simplifying inequalities is an essential skill in algebra. It involves breaking down complex expressions into simpler forms to better understand their relationships. This can include combining like terms, performing arithmetic operations, and isolating variables.

In this exercise, we simplified the inequalities by substituting the specific point and performing arithmetic operations. For instance, in inequality (d) \(y \leq -\frac{1}{2}x + 3\), substitute \(x = -1\) and \(y = 3\), the inequality becomes \(3 \leq - -0.5 + 3\), simplifying further to \(3 \leq 3.5\), which is true.

This process of simplification helps students focus on understanding the core relationships within an inequality, making it easier to determine if a point satisfies it.

Coordinate geometry is the study of geometric figures through a coordinate system. It helps visualize algebraic equations and inequalities, making it easier to determine if points lie within certain regions.

In this exercise, students are asked to determine if the point (-1, 3) satisfies various inequalities. Visualizing the inequalities on a coordinate plane helps understand why certain points do or do not satisfy them.

For instance, consider inequality (b) \(x < 3\). On a coordinate plane, this is all the area to the left of the vertical line \(x = 3\). The point (-1, 3) clearly lies within this region, thus satisfying the inequality.

Using coordinate geometry not only aids in solving inequalities but also enhances spatial reasoning, which is valuable in various fields of science, technology, engineering, and mathematics (STEM).